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## Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

#### Tayfun TANBAY [1]

The meshless radial basis function collocation method is an efficient numerical technique for solving partial differential equations. The multiquadric is the most widely utilized radial function for this purpose; but it contains a shape parameter, which has a significant effect on the performance of the method. In this study, the meshless collocation method employing multiquadric as the radial function with optimum shape parameters is applied to the numerical solution of the multigroup neutron diffusion equation. The optimization of the shape parameter is performed by minimizing the Madych-Nelson function. One external and two fission source problems are solved to investigate the performance of the method. The results show that the meshless collocation method with optimized shape parameters yield a high level of accuracy with an exponential convergence rate.

Meshless, RBF, Collocation, Optimum shape parameter, Neutron diffusion
• [1] Kansa, E.J. (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics, In: R. Crosbie, P. Luker (Eds.), Proceedings of the 1986 Summer Computer Simulation Conference, San Diego, Society for Computer Simulation, 4:111-117.
• [2] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
• [3] Hardy, R.L. (1971). Multiquadratic equations for topography and other irregular surfaces. Journal of Geophysical Research, 76:1905-1915.
• [4] Franke, R. (1982). Scattered data interpolation tests of some methods. Mathematics of Computation, 38:181-200.
• [5] Fasshauer, G.E. (2002). Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43:423-438.
• [6] Rippa, S. (1999). An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, 11:193-210.
• [7] Esmaeilbeigi, M. and Hosseini, M.M. (2014). A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method. Applied Mathematics and Computation, 249:419-428.
• [8] Koupaei, J.A., Firouznia, M. and Hosseini, M.M. (2018). Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alexandria Engineering Journal, 57:3641-3652.
• [9] Luh, L.T. (2019). The choice of the shape parameter-A friendly approach. Engineering Analysis with Boundary Elements, 98:103-109.
• [10] Tanbay, T. and Ozgener, B. (2013). Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Mathematical and Computational Applications, 18:399-407.
• [11] Tanbay, T. and Ozgener, B. (2014). A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations. Engineering Analysis with Boundary Elements, 46:30-40.
• [12] Tanbay, T. (2018). On the accuracy and stability of the meshless RBF collocation method for neutron diffusion calculations. Journal of Innovative Science and Engineering, 2:8-18.
• [13] Fedoseyev, A.I., Friedman, M.J. and Kansa, E.J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications, 43:439-455.
Birincil Dil en Mühendislik Research Articles Orcid: 0000-0002-0428-3197Yazar: Tayfun TANBAY (Sorumlu Yazar)Kurum: BURSA TECHNICAL UNIVERSITYÜlke: Turkey Yayımlanma Tarihi : 24 Haziran 2019
 Bibtex @araştırma makalesi { jise570328, journal = {Journal of Innovative Science and Engineering}, issn = {}, eissn = {2602-4217}, address = {ursa Technical University, Mimar Sinan Campus, Mimar Sinan Mah. Mimar Sinan Blv. Eflak Cad. No:177 16310 Yıldırım, Bursa / Turkey}, publisher = {Bursa Teknik Üniversitesi}, year = {2019}, volume = {3}, pages = {23 - 31}, doi = {10.38088/jise.570328}, title = {Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters}, key = {cite}, author = {Tanbay, Tayfun} } APA Tanbay, T . (2019). Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters . Journal of Innovative Science and Engineering , 3 (1) , 23-31 . DOI: 10.38088/jise.570328 MLA Tanbay, T . "Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters" . Journal of Innovative Science and Engineering 3 (2019 ): 23-31 Chicago Tanbay, T . "Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters". Journal of Innovative Science and Engineering 3 (2019 ): 23-31 RIS TY - JOUR T1 - Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters AU - Tayfun Tanbay Y1 - 2019 PY - 2019 N1 - doi: 10.38088/jise.570328 DO - 10.38088/jise.570328 T2 - Journal of Innovative Science and Engineering JF - Journal JO - JOR SP - 23 EP - 31 VL - 3 IS - 1 SN - -2602-4217 M3 - doi: 10.38088/jise.570328 UR - https://doi.org/10.38088/jise.570328 Y2 - 2019 ER - EndNote %0 Journal of Innovative Science and Engineering Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters %A Tayfun Tanbay %T Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters %D 2019 %J Journal of Innovative Science and Engineering %P -2602-4217 %V 3 %N 1 %R doi: 10.38088/jise.570328 %U 10.38088/jise.570328 ISNAD Tanbay, Tayfun . "Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters". Journal of Innovative Science and Engineering 3 / 1 (Haziran 2019): 23-31 . https://doi.org/10.38088/jise.570328 AMA Tanbay T . Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019; 3(1): 23-31. Vancouver Tanbay T . Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering. 2019; 3(1): 23-31. IEEE T. Tanbay , "Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters", Journal of Innovative Science and Engineering, c. 3, sayı. 1, ss. 23-31, Haz. 2019, doi:10.38088/jise.570328

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